528 
Mr Young, On Monotone Sequences 
sequence of continuous and, in general, unbounded functions 
<j>i, (f} 2 , — If </>», be unbounded, express it as the limit of a 
monotone ascending uniformly convergent, or divergent, sequence 
of bounded continuous functions <£ n>1 , < p n>2 , Then the upper 
integral of f j f(x)dx, is defined by the following equations, 
I / (x) dx — Lt / (p n (x) dx, 
J 71— CO J 
The lower integral of f, j f(x) dx, being similarly defined, we 
have a Riemann integral when, and only when, the upper and 
lower integrals have the same value for the interval, or region, of 
integration considered. 
It will be observed that the integration is reduced to that of 
a bounded continuous function f, which may be supposed defined 
in any of the various ways which is convenient, e.g. in the case of 
a single independent variable, as the inverse differential coefficient 
of the integrand. 
Assuming, further, that such a definition has been given for 
the integration of a bounded continuous function with respect to 
a closed set of points, the above gives us a definition of the integral 
of any function with respect to such a set. 
The definition arises naturally from the application of the 
principle of the conservation of mathematical laws. In my 
“ General Theory of Integration J,” indeed, I proved the funda- 
mental proposition that, in the case of bounded functions, the 
upper integral of a function is equal to that of its associated 
upper limiting function. This proposition, applied now to un- 
bounded functions, appears in the above as itself a definition of 
an improper integral. 
On the other hand, as unbounded and discontinuous functions 
present themselves most naturally as the limits of bounded con- 
tinuous functions, it is a natural extension of the meaning of the 
* It is necessary of course to shew that these limits which from the very nature 
of the sequences exist, are independent of the particular mode in which the con- 
stituent functions of the sequences are chosen so long as they are monotone. This 
presents no great difficulty. 
f Which may be taken to be a polynomial. In fact by an extension of a 
theorem of Weierstrass anj' unbounded continuous function may be expressed as 
the uniform limit of a polynomial. I have given a proof of this theorem in a 
paper recently presented to the London Mathematical Society. Thus the problem 
of integration is reduced to that of integrating a positive integral power of x. 
+ Phil. Trans. Vol. 204 (1905). 
